Kinematics of Machines. 485 Revolutions of mandrel _ No. of threads perjnch. on mandrel Revolutions of leadingscrew ~ No. of threads per iF. oiileadingscSrew in order to cut a definite pitch. This may also be stated as followers at L. S.jsnd _ pitch L. S. drivers at M end ~~ pitch M screw or the pitches and wheels are in the same ratio^ which ratio, being found, must be accommodated by a suitable train. Example 46. — In Plate V., the leading screw being. £" pitch, and the wheels in the set rising by 5 at a time from 20 to 120 teeth, it is required to arrange wheels to cut (i) a screw of ro threads per inch, right-handed, and (2) a screw of L" pitch left-handed. Pitch ratio - Putting 30 teeth on n (Fig. 135) into 75 on stud (£, Fig. 140): 30 teeth on stud into 90 on L.S., we have, 75 x 9° ^ i§ and the handle at n must be down. wheel ratio = (2) pitch ratio 30 x 30 *±§- - 1 M"" ~ i = Putting 45 teeth on L.S. and 60 teeth on n; with any intermediate on stud (say 60) we have, wheel ratio - 11 _ 3 and the h^nd1^ at n must ----------- 60 4 btup. Kinematics (of Machines) is a method of attacking machine problems devised by Prof. Reuleaux, and anglicised by Prof. Kennedy. We shall proceed to discuss its principles. Pairs.—The constraining parts are termed pairs because they always occur in sets of two. Of these there are higher and lower pairs. The former connect by points or lines, but the latter by their whole surfaces. Three kinds of lower pairs are possible: I. Sliding pairs, as a piston and cylinder. II. Turning pairs, as a journal or pin. III. Screw pairs, including all screws and nuts. Complete or dosed pairs have their motions fully defined: incomplete pairs require further closure, as at Fig. 444, where gravity is not for the moment considered.